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Statistics
Advanced Regression
Association
Box plot
Central Limit Theorem
Confidence interval
Correlation coefficient
Covariance
Empirical rule
Estimation
Expected value
Measures of Position
Median
Mode
Normal distribution
Regression
Sampling distributions
Significance Level
Significance Test
Standard Deviation
Standard Error
Variable types
Variance
Z-score

Sampling distributions

When people enter an apple store, of them buys a product before leaving(this is the real proportion, it is the ground truth and we have it).

Imagine we sample the population and try to obtain p from the samples.
Now P becomes uncertain, it is random variable :

According to the Central Limit Theorem, for large samples, the sample proportion is approximately normally distributed, with mean:
and standard deviation of a proportion:

Where:

  • is the proportion/statistic of something.
  • is the mean of the distribution of sampled proportions.
  • is the standard deviation of the sampled proportions.

If we are investingating a mean(not a proportion) the formula for standard deviation is:

Warning

Sometimes we want to compute the probability of successes being more than a certain number.

We know that we can get the area under a curve by using the z-scores, but this distribution only approximates a normal distribution when using a large n.

So when we have a small n we need to go sideways:

  • If our configuration is also a binomial distribution, we can use that formula to compute every single discrete probability.
Example

For the population of individuals who own an iPhone, suppose p = 0.25 is the proportion that has a given app.

  1. For a random sample of size n = 4, and the mean and the standard deviation of the sampling distribution of the sample proportion:

  1. Find the probability that the proportion of having the app is at least 0.75 when n = 4.

    Here the sample size is too small, so we can't use the normal distribution stuff.
    0.75 of 4 = 3, so we need the probability that at least 3 people have the app.
    We do that by summing the probabilities that 3 people have the app and 4 people have the app.
    Since those probabilities are discrete and there are only 2 possible outcomes per trial, we can use the binomial distribution formula

Example

In the population, IQ scores are normally distributed with mean µ = 100 and variance σ 2 = 15. Suppose to draw a random samples of 25 individuals from the population and measure the IQ score

  1. Compute the probability of observing a sample mean between 98 and 102 when drawing a sample of 25 individuals:

The standard deviation of the sample mean for a sample of size 25. Is given by:

The z-scores for 98 and 102 are:

The areas given by the z-tables for the z-scores are:

The area between the two z-scores is given by:

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Sampling distributions